Fast Type-II Hartley Transform Algorithms for Short-Length Input Sequences

Abstract

This paper presents the type-II fast discrete Hartley transform (DHT-II) algorithms for input data sequences of lengths from 2 to 8. The starting point for developing the eight algorithms is the representation of DHT-II as a matrix–vector product. The underlying matrices usually have a good block structure. These matrices must then be successfully factorized to obtain a computational procedure that reduces the number of operations in computing the matrix–vector product. In some cases, it is necessary to pre-decompose the original matrices into submatrices and rearrange the rows and/or columns of the resulting matrices to find the factorizations that would substantially save the arithmetic operations. As a result of applying the pointed transformations, we synthesized the final algorithms with reduced computational complexity. The correctness of the obtained algorithmic solutions was theoretically justified using the rigorous mathematical background of each of them. Then, the complex algorithms were further tested using the MATLAB R2023b software to confirm their performance. Finally, an evaluation of the computational complexity for each obtained solution was compared with the computational complexity of the direct calculation of the matrix–vector product and existing fast DHT-II algorithms. The obtained factorizations of the DHT-II transformation matrices on average reduce the number of additions by 5% and the number of multiplications by 73% compared with the direct calculation of the matrix–vector product.